Let's move the discussion back to the list...
On 24/01/2008, Hugo Coolens <hugocoolens at skynet.be> wrote:
> The thing is, I have a whole family of polynomials related to an
> electrical network and I'd love to see a general method to solve
> them (factorize or determine the roots) starting from the coefficients,
> I don't know much about Galois theory but it would be great just to see
> it work on an example like I sent you, maybe then I'll get the necessary
> insight
This doesn't seem like an easy question in general. Of course if you
can find enough roots by radicals until your polynomial is of degree 4
or less, then you have found them all. In general, I think we would
first need routines for computing Galois groups of polynomials of
small degrees. Last time I checked, Maple had routines for computing
Galois groups of polynomials of degree up to 9, but I don't know if
Maxima has any such thing.
Using tricks such as computing discriminants and other assorted
methods, it's possible to automate the computation of the Galois group
for polynomials. A decomposition series for the Galois group, if it
exists (i.e. if it's soluble), can then give a clue on what radicals
the solution will be, although I don't have details for this.
I would have to grep the literature to see if I can find anything
about computer algorithms for solutions by radicals or determine that
they're impossible. It's an interesting problem, and I'm sure someone
else has already solved it or worked on it extensively.
Anyone else, please feel free to contribute further thoughts.
- Jordi G. H.